Abstract Nonsense

Crushing one theorem at a time

A Classification of C-representations With No (rho,J)-invariant Subspaces


Point of post: In this post we shall classify, up to equivalence, the set of all \mathbb{C}-representations of a finite group G. This will enable us, due to a previous characterization, the set of all irreducible \mathbb{R}-characters of a finite group G in terms of the irreducible \mathbb{C}-characters.

 

Motivation

Clearly our goal in these last few posts was to see how much information about irreducible \mathbb{R}-representations we could glean from our knowledge of irreducible \mathbb{C}-representations in the sense that if we know everything we could possibly want to know about the latter for a specific group what could we say about the former. We began this search by showing that there was a bijection between equivalency classes of \mathbb{C}-representations \rho satisfying the real condition with realizer J and no non-trivial proper \left(\rho,J\right)-invariant subspaces and equivalency classes of  irreducible \mathbb{R}-representations where the class [\rho] in the domain was mapped to the class [\rho_\Re] in the codomain. We then showed that there was a natural way to create these special \mathbb{C}-representations from irreducible representations which were either complex or quaternionic. What we shall now do in this post is classify all of these special \mathbb{C}-representations by showing that, up to equivalence, any such one is either a real irreducible \mathbb{C}-representation or of the form of one of the representations created by a complex or quaternionic irrep. This will allow us then, since characters are invariant under equivalence, to ascertain all the irreducible \mathbb{R}-characters of the group in terms of the irreducible \mathbb{C}-characters.

 

Classification of \mathbb{C}-representations Satisfying the Real Condition With Realizer J and having no non-trivial proper \left(\rho,J\right)-invariant Subspaces

For the proof that follows allow \Delta_G to be the set of all \mathbb{C}-representations on G satisfying the real condition where, if the realizer is J, there does not exist any non-trivial proper \left(\rho,J\right)-invariant subspaces. With this in mind:

 

Theorem: Let G be a finite group and \rho\in\Delta_G with realizer J. Then, either \rho is a real irreducible \mathbb{C}-representation or there exists a complex or quaternionic irrep \psi and a complex conjugate K such that \rho\cong\psi\oplus\text{Conj}_\psi^K ( in accordance with the construction in the last theorem).

Proof: Since being a representation of G which satisfies the real condition with no non-trivial proper (\rho,J)-invariant subspaces we may then assume that \rho is the representation \displaystyle \psi_1\oplus\cdots\oplus\psi_n:G\to\mathcal{U}\left(\mathscr{V}_1\oplus\cdots\mathscr{V}_n\right) where \psi_k:G\to\mathcal{U}\left(\mathscr{V}_k\right) is an irrep for k=1,\cdots,n and K the complex conjugate on \mathscr{V}_1\oplus\cdots\oplus\mathscr{V}_n which is realizer for \psi_1\oplus\cdots\oplus\psi_n. Now, if n=1 then \psi_1\oplus\cdots\oplus\psi_n=\psi_1 and \psi_1 is an irrep which must be real since it must commute with K (of course we are appealing to our previous characterization of self-conjugate irreps). Now, suppose that n\geqslant 2. We claim that if \mathscr{H}=\mathscr{V}_1\times\{\bold{0}\}\times\cdots then \mathscr{H}+J\left(\mathscr{H}\right) is a \left(\psi_1\oplus\cdots\oplus\psi_n,J\right)-invariant subspace. Indeed, this is clear, namely if (h,0,\cdots)+J((h',0,\cdots))\in\mathscr{H}+J\left(\mathscr{H}\right) then for every g\in G

 

\displaystyle \begin{aligned}\left(\psi_1(g)\oplus\psi_n(g)\right)\left((h,0,\cdots)+J\left((h',0,\cdots)\right)\right) &=\left(\left(\psi_1(g)\right)(h),0,\cdots\right)+J\left(\psi_1(g)\oplus\cdots\oplus\psi_n(g)\right)\left(h',0,\cdots,\right)\\ &= \left(\left(\psi_1(g)\right)(h),0,\cdots,\right)+J\left(\left(\psi_(g)\right)(h'),\cdots\right)\end{aligned}

 

and this last term is clearly in \mathscr{H}_1+J\left(\mathscr{H}_1\right). Since g\in G and h+J(h')\in\mathscr{H} were arbitrary it follows that \mathscr{H}+J\left(\mathscr{H}\right) is \psi_1\oplus\cdots\psi_n-invariatn as claimed, and since \mathscr{H}+J\left(\mathscr{H}\right) is evidently J-invariant, \left(\psi_1\oplus\cdots\oplus\psi_n,J\right)-invariance follows. But, by assumption this implies that \mathscr{V}_1\oplus\cdots\oplus\mathscr{V}_n=\mathscr{H}+J\left(\mathscr{H}\right) from where it follows that n=1 if J\left(\mathscr{H}\right)=\mathscr{H} and n=2 otherwise. So, we are restricting our attention to representations of the form \psi_1\oplus\psi_2:G\to\mathcal{U}\left(\mathscr{V}_1\oplus\mathscr{V}_2\right). Let, \mathscr{H}_1=\mathscr{V}_1\times\{\bold{0}\} and \mathscr{H}_2=\{\bold{0}\}\times\mathscr{V}_2 and \pi_1,\pi_2 the projections on these subspaces respectively. Note from the above we’ve determined that J\left(\mathscr{H}_1\right)\ne\mathscr{H}_1. Consider then the mapping \pi_2J\pi_1:\mathscr{V}_1\oplus\mathscr{V}_2\to\mathscr{H}_2. This clearly must be a non-zero transformation since there exists some (v,0)\in\mathscr{H}_1 such that J(v,0)\notin\mathscr{H}_1, and so J(v,0)=(u,w) where w\ne 0 and so \pi_2J\pi_1(v,0)=\pi_2J(v,0)=\pi_2(u,w)=w\ne 0. Moreover, there is an interesting relationship between the map \pi_2J\pi_1\left(\psi_1(g)\oplus\mathbf{1}\right) for each g\in G. Namely, for any (u,v)\in\mathscr{V}_1\oplus\mathscr{V}_2

 

\begin{aligned}\pi_2J\pi_1\left(\psi_1(g)\oplus\mathbf{1}\right)(u,v) &=\pi_2J\pi_1\left(\left(\psi_1(g)\right)(u),v\right)\\ &=\pi_2J\left(\left(\left(\psi_1(g)\right)(u),0\right)\right)\\ &=\pi_2J\left(\psi_1(g)\oplus\psi_2(g)\right)(u,0)=\pi_2\left(\psi_1(g)\oplus\psi_2(g)\right)J(u,0)\\ &=\left(\mathbf{1}\oplus\psi_2(g)\right)\pi_2 J(u,0)\\ &= \left(\mathbf{1}\oplus\psi_2(g)\right)\pi_2 J\pi_1(u,v)\end{aligned}

 

From this, it follows that if C is a complex conjugate on \mathscr{V}_2 then we have

 

\left(\mathbf{1}\oplus C\psi_2(g)C\right)\left(\mathbf{1}\oplus C\right)\pi_2 J\pi_1=\left(\mathbf{1}\oplus C\right)\pi_2 J\pi_1\psi_1(g)

 

Thus, if T:\mathscr{V}_1\to\mathscr{V}_2 is given by v\mapsto (v,0)\mapsto \left(\mathbf{1}\oplus C\right)\pi_2J\pi_1=(u,w)\mapsto w then T is an intertwinor for \psi_1 and \text{Conj}_{\psi_2}^{C} and since T is non-zero by prior discussion we may conclude by Schur’s lemma that \psi_1\cong\text{Conj}_{\psi_2}^C. Thus, \psi_1\oplus\psi_2\cong\psi_1\oplus \text{Conj}_C^{\psi_1}. The conclusion follows. \blacksquare

 

 

 

References:

1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

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April 4, 2011 - Posted by | Algebra, Representation Theory | , , ,

1 Comment »

  1. […] We know from our result concerning the bijection of with that there exists some so that . But, we know that for each one has that is equivalent to where is some real irreducible -representation, […]

    Pingback by Representation Theory: Irreducible R-characters of a Finite Group in Terms of its Irreducible C-characters « Abstract Nonsense | April 4, 2011 | Reply


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